Properties

Label 122034f
Number of curves 4
Conductor 122034
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("122034.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 122034f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
122034.g3 122034f1 [1, 1, 1, -10208, -378691] [2] 314496 \(\Gamma_0(N)\)-optimal
122034.g4 122034f2 [1, 1, 1, 8282, -1576843] [2] 628992  
122034.g1 122034f3 [1, 1, 1, -148883, 21964625] [2] 943488  
122034.g2 122034f4 [1, 1, 1, -74923, 43886369] [2] 1886976  

Rank

sage: E.rank()
 

The elliptic curves in class 122034f have rank \(1\).

Modular form 122034.2.a.g

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} - q^{6} - 2q^{7} + q^{8} + q^{9} - q^{11} - q^{12} - 4q^{13} - 2q^{14} + q^{16} - 6q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.